Optimization of One-Way Quantum Computation Measurement Patterns

In one-way quantum computation (1WQC), an initial highly entangled state, called a graph state, is used to perform universal quantum computations by a sequence of adaptive single-qubit measurements and post-measurement Pauli-X and Pauli-Z corrections. 1WQC computation can be represented by a measurement pattern (or simply a pattern). The entanglement operations in a pattern can be shown by a graph which together with the identified set of its input and output qubits is called the geometry of the pattern. Since a pattern is based on quantum measurements, which are fundamentally nondeterministic evolutions, there must be conditions over geometries to guarantee determinism. These conditions are formalized by the notions of flow and generalized flow (gflow). Previously, three optimization methods have been proposed to optimize 1WQC patterns which can be performed using the measurement calculus formalism by rewriting rules. However, the serial implementation of these rules is time consuming due to executing many ineffective commutation rules. To overcome this problem, in this paper, a new scheme is proposed to perform the optimization techniques simultaneously on patterns with flow and only gflow based on their geometries. Furthermore, the proposed scheme obtains the maximally delayed gflow order for geometries with flow. It is shown that the time complexity of the proposed approach is improved over the previous ones.     

Homomorphisms and chains of Kripke models

In this paper we define a suitable version of the notion of homomorphism for Kripke models of intuitionistic first-order logic and characterize theories that are preserved under images and also those that are preserved under inverse images of homomorphisms. Moreover, we define a notion of union of chain for Kripke models and define a class of formulas that is preserved in unions of chains. We also define similar classes of formulas and investigate their behavior in Kripke models. An application to intuitionistic first-order arithmetic is also given.     

A Complete Solution of the Wave Equations for Transversely Isotropic Media

A transversely isotropic material in the sense of Green is considered. A complete solution in terms of retarded potential functions for the wave equations in transversely isotropic media is presented. In this paper we reduce the number of potential functions to only one, and we discuss the required conditions. As a special case, the torsionless and rotationally symmetric configuration with respect to the axis of symmetry of the material is discussed. The limiting case of elastostatics is cited, where the solution is reduced to the Lekhnitskii–Hu–Nowacki solution. The solution is simplified for the special case of isotropy. In this way, a new series of potential functions (to the best knowledge of the author) for the elastodynamics problem of isotropic materials is presented This solution is reduced to a special case of the Cauchy–Kovalevski–Somigliana solution, if the displacements satisfy specific conditions. Finally, Boggio's Theorem is generalized for transversely isotropic media which may be of interest to the reader beyond the present application. Dedicated to Morton E. Gurtin     

Elastodynamics and Elastostatics by a Unified Method of Potentials for x 3-Convex Domains

A new general solution in terms of two scalar potential functions for classical elastodynamics of x ₃-convex domains is presented. Through the establishment and usage of a set of basic mathematical lemmas, a demonstration of its connection to Kovalevshi–Iacovache–Somigliana elastodynamic solution, and thus its completeness, is realized with the aid of the theory of repeated wave equations and Boggio’s theorem. With the time dependence of the potentials suppressed, the new decomposition can, unlike Lamé’s, degenerate to a complete solution for elastostatic problems.      

A capillary force model for interactions between two spheres

Based on the method of energy principle, an analytical approach for computing the capillary force for sphere/sphere geometry is presented in this paper. In modeling the capillary force, we consider spheres with both equal and non-equal radii, for both symmetric and asymmetric configurations at liquid/solid interfaces. We use numerical analysis to investigate the validity and efficiency of the derived model. The effect of various parameters including humidity, distance between two spheres, radii of spheres and contact angles on the meniscus force are investigated. Finally the results obtained from the model are compared with experimental measurements, and the accuracy and precision of the presented approach is verified.